Quasi-hidden Markov model and its applications in change-point problems

In a hidden Markov model (HMM), the observed data are modelled as a Markov chain plus independent noises, hence loosely speaking, the model has a short memory. In this article, we introduce a broad class of models, quasi-hidden Markov models (QHMMs), which incorporate long memory in the models. We develop the forward-backward algorithm and the Viterbi algorithm associated with a QHMM. We illustrate the applications of the QHMM with the change-point problems. The structure of the QHMM enables a non-Bayesian approach. The input parameters of the model are estimated by the maximum likelihood principle. The exact inferences on change-point problems under a QHMM have a computational cost O(T-2), which becomes prohibitive for large data sets. Hence, we also propose approximate algorithms, which are of O(T) complexity, by keeping a long but selected memory in the computation. We illustrate with step functions with Gaussian noises and Poisson processes with changing intensity. The approach bypasses model selection, and our numerical study shows that its performance is comparable and sometimes superior to the binary segmentation algorithm and the pruned exact linear time method.